Sharp control time for viscoelastic bodys

TL;DR

This paper demonstrates that viscoelastic bodies can be controlled in the same sharp time as their memoryless telegraph counterparts by comparing their equations and employing moment method techniques.

Contribution

It establishes the exact controllability of viscoelastic systems in the same minimal time as the telegraph equation using a novel comparison and Riesz sequence analysis.

Findings

Viscoelastic systems are controllable at the same times as telegraph equations.

A moment method approach proves the Riesz sequence property.

Results extend existing controllability theories for viscoelastic bodies.

Abstract

It is now well understood that equations of viscoelasticity can be seen as perturbation of wave type equations. This observation can be exploited in several different ways and it turns out that it is a usefull tool when studying controllability. Here we compare a viscoelastic system which fills a surface of a solid region (the string case has already been studied) with its memoryless counterpart (which is a generalized telegraph equation) in order to prove exact controllability of the viscoelastic body at precisely the same times at which the telegraph equation is controllable. The comparison is done using a moment method approach to controllability and we prove, using the perturbations theorems of Paley-Wiener and Bari, that a new sequence derived from the viscoelastic system is a Riesz sequence, a fact that implies controllability of the viscoelastic system. The results so obtained generalize existing controllability results and furthermore show that the ``sharp'' control time for the telegraph equation and the viscoelastic system coincide.